Noise suppression via generalized-Markovian processes

It is by now well established that noise itself can be useful for performing quantum information processing tasks. We present results which show how one can effectively reduce the error rate associated with a noisy quantum channel by counteracting its detrimental effects with another form of noise. In particular, we consider the effect of adding on top of a purely Markovian (Lindblad) dynamics, a more general form of dissipation, which we refer to as generalized-Markovian noise. This noise has an associated memory kernel and the resulting dynamics are described by an integrodifferential equation. The overall dynamics are characterized by decay rates which depend not only on the original dissipative time scales but also on the new integral kernel. We find that one can engineer this kernel such that the overall rate of decay is lowered by the addition of this noise term. We illustrate this technique for the case where the bare noise is described by a dephasing Pauli channel. We analytically solve this model and show that one can effectively double (or even triple) the length of the channel, while achieving the same fidelity, entanglement, and error threshold. We numerically verify this scheme can also be used to protect against thermal Markovian noise (at nonzero temperature), which models spontaneous emission and excitation processes. A physical interpretation of this scheme is discussed, whereby the added generalized-Markovian noise causes the system to become periodically decoupled from the background Markovian noise.

Figure 1

Minimum channel fidelity $f\left(t\right)=\frac{1}{2}[1+\mathrm{\Lambda }\left(t\right)]$, as a function of time for the purely Markov (background channel) evolution [blue (dashed), $B=0]$ and for the combined system aided by the generalized-Markovian noise [red (solid), $B>0]$. We also plot the fidelity of the background channel, with $\gamma$ rescaled to $\frac{1}{2}(\gamma +1/2{\tau }_k)$ [yellow (dot-dashed)], which intersects the $B>0$ curve at times $t=\frac{2}{\omega }(\pi n-\varphi )$ and $2\pi n/\omega$. Here $\gamma =1,{\tau }_k=25$, and we set $B$ such that $\omega T=2\pi$ (for $T=1$). Note that the axis of dephasing is not important here; $f$ is the same for each direction (for fixed parameters). Time is measured in units of $1/\gamma$.

Figure 2

Difference in the fidelity $\mathrm{\Delta }F:=F-F_0$ for all pure states (single qubit), where $F_0$ is the fidelity of the “background” channel, and $F$ the fidelity the “combined” channel, at time $t=T$ (cf. Fig. 1). The background channel here corresponds to dephasing in the $x$ direction. Left: $\theta ,\varphi$ axes correspond to positions on the Bloch sphere for a single qubit: $|\psi \rangle =cos(\theta /2)|0\rangle +e^{i\varphi }sin(\theta /2)|1\rangle$. Right: Spherical plot of $\mathrm{\Delta }F$ (i.e., on the Bloch sphere). We see for all nonstationary initial states, the fidelity increases. Parameters: $T=1,\gamma =1$ (hence $p_0\approx 43\%$), and ${\tau }_k=25$, with $B$ chosen so that $\omega T=2\pi$ (hence $p\approx 32\%$).

Figure 3

Concurrence as a function of time, for an initially maximally entangled pure 2-qubit state, $|\psi \rangle =\frac{1}{\sqrt{2}}\left(\right|00\rangle +|11\rangle )$. Here the dephasing is in the $z$ direction acting on one of the qubits. One can show $\mathcal{C}\left(t\right)=\left|\mathrm{\Lambda }\right(t\left)\right|$. We plot for both the background channel [blue (dashed)], and for the channel assisted by the generalized-Markovian noise [red (solid)]. We see the peaks for the assisted channel surpass the background channel. Here $\gamma =1$, $B=5$, ${\tau }_k=5$. Time is measured in units of $1/\gamma$.

Figure 4

Minimum channel fidelity against time for cosine memory kernel. We plot for the background channel (blue/dash, $B=0$), the combined channel (red/solid, $B>0$), and for the background channel with $\gamma$ replaced by $\frac{1}{3}(\gamma +1/{\tau }_k)$ (yellow/dot-dash). Here $\gamma =0.5,\nu =10,{\tau }_k=25$. For this parameter choice, the peaks occur approximately at the times $2\pi n/\omega$. We numerically verify the generated map is CP for all $t\ge 0$ (including the case when we set $\gamma =0$). Time is measured in units of $1/\gamma$.

Figure 5

Fidelity overlap with initial pure state ${\rho }_0=|{\psi }_0\rangle \langle {\psi }_0|$ as a function of time for thermal qubit, with [red (solid), ${\gamma }_z=2]$ and without [blue (dashed), ${\gamma }_z=0]$ added protection by generalized Markovian noise. Parameters: ${\gamma }_{-}=1,{\gamma }_{+}=1/2,{\tau }_k=5$. Initial state: $|{\psi }_0\rangle =\frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$. We numerically verify the dynamics are CP (using the Choi matrix). Inset: Distance of the state at time $t$ to the corresponding steady state of ${\mathcal{L}}_0$. We see periodically the system passes close to the steady state (when ${\gamma }_z\ne 0$). When the distance $\parallel ({\mathrm{\Phi }}_t-P_0){\rho }_0\parallel \approx 0$, the system is essentially decoupled from the thermal noise. We use the maximum singular value norm. Time is measured in units of $1/{\gamma }_{-}$.